How Much Should a 19th-Century French Bastard Inherit
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چکیده
Catalan's formula, for the portion of the inheritance that a legitimate child of a 19th-century deceased French gentleman should receive, is given a new proof (using Difference Operators), and generalized. Another, more computationally efficient, formula is also derived. Catalan's Formula Chrystal's classical text[Ch] (II, p. 416) contains the following charming exercise, due to Cata-lan[Ca]. By French law an illegitimate child receives one-third of the portion of the inheritance that he would have received had he been legitimate. If there are l legitimate and n illegitimate children, show that the portion of inheritance 1 due to a legitimate child is 1 l − n 3l(l + 1) + n(n − 1) 3 2 l(l + 1)(l + 2) Let's call the portion due to a legitimate child a(l, n). If the condition would have been that each illegitimate child receives one-third of what a legitimate child is to receive, this would have yielded the simple algebraic equation la(l, n) + n(1/3)a(l, n) = 1 and hence a(l, n) = 3/(3l + n). But since the bastard is to receive one-third of what he would have received had he been legitimate, we have a difference (or recurrence) equation: la(l, n) + (n/3)a(l + 1, n − 1) = 1 .
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